Faulty Premise/Argument/Assertion or Something Else?

So many fallacies so little time. This is nothing more than political spin.

Ohhhh lord. As if there hasn't been enough Israel-Palestine derangement already. The wider 'conversation' on this topic, as with other sociopolitical trends like BLM, MeToo etc, is littered with bad arguments peddled by pseudoexperts and strong feelings taken as fact. No wonder you end up with sentiments such as the ones expressed in the above links.

To summarise, it's typically a mixture of fallacies and questionable/false premises, and a wider failure to exercise cognitive empathy - yes, on both sides.

The media often calls the fighting between Israel and Palestine a 'conflict'. 

Not just the media; the general public refers to it as a conflict as well. A violent dispute between two opposing parties is typically referred to as some sort of conflict. This part isn't fallacious, but the person behind it might be using the phrase 'the media' to try and pre-emptively discredit the notion that it is a conflict.

Conflict, as stated, implies there are two equal sides to this 'conflict'.

Nonsense. A conflict requires two or more disparate sides. Nowhere is it stated that they have to be 'equal', since in at least one sense, one side will be more powerful than another, at least slightly so - e.g. having more supporters, more economic resources, etc. That also sheds light on the vague concept of 'equality' in this premise.

According to this word game, the U.S. Civil War wasn't a conflict because the Union was stronger than the Confederacy. The Punic Wars weren't conflicts, because Rome was stronger than Carthage.

P3: This implies that the Israelis and the Palestinians are on a equal footing with one another.

P2 is false, and thus no longer implies P3 (inference objection). There is no need for equality of military strength between the opposite sides.

C1: In Israel and Palestine, the two sides are not equal in military power. The Palestinians don't even have an proper military. Israel, meanwhile, has a powerful military backed by the US and western nations.

"Don't even have a proper military" is questionable. However, even if you believe it, it is irrelevant. Firstly, Israel is not using all of its military might at any one time. Thus, in practical terms, the difference between the two is not as dramatic as the person is making out. Secondly, as mentioned above, 'conflict' just isn't defined like this. The person is changing the definition to suit their argument (definist fallacy).

C2: The use of overwhelming firepower by one side against another side that has little firepower is not equal.

It may be true that Israel has more 'firepower' than Palestine. However, 'overwhelming' implies some sort of nefarious/oppressive intent on Israel's part, which has not been supported with evidence. They would claim they are using firepower defensively. The Palestinians would claim otherwise.

The one without power (the Palestinians), the oppressed, cannot be in a 'conflict' with the ones with power, the oppressor (the Israelis) because there is no two equal sides to the conflict. The colonized cannot be in 'conflict' with its coloniser because of this unequal power dynamic.

We jump from two sides that are admittedly unequal in terms of firepower to a 'coloniser-colonised' dynamic (begging the question). Why is Israel the 'coloniser' and 'oppressor'? Did the Union 'colonise' the Confederacy?

Based on how the two sides are not equal and how there is the oppressor (the Israelis) and the oppressed (the Palestinians), the Israel Palestinian 'conflict' isn't actually a conflict at all. It is a genocide/ethnic cleansing/military occupation and anyone who uses the word 'conflict' is complicit in Israeli apologia.

We did not ascertain that there was an oppressor and an oppressed. You made that part up to suit the narrative you're trying to peddle. Lastly we have an unsupported claim regarding 'genocide/ethnic cleansing', and some prejudicial language (accusing people of being 'complicit' in Israeli propaganda).